Is there a way to combine/flatten multiple edges together that are in a checkerboard pattern?
If I'm understanding your question correctly, you should be able to select the 2 planes and Join so they are the one mesh. Then if you select the vertices you want merged you can click Remove Doubles and it will remove the duplicate vertices, essentially merging them together.

EDIT: Now that I understand what you were meaning, the only suggestion I have is to get rid of the vertical edges and create them within the original mesh. I tried it myself and it's a little tedious, but gets the same result I think you're after. First subdivide the bottom edge into how many segments you want, but remember that the Number of Cuts is one less than the number of segments (eg. 9 cuts for 10 segments) and then manually connect each set of vertices and pressing J. If that doesn't achieve the results you're after then I'm afraid I'm out of ideas :)